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Chapter 3: Problem 28

In uniform circular motion, which of the following are correcl? (a) \(|\Delta \vec{v}| \neq 0\) but \(\Delta|\vec{v}|-0\) (b) \(|\Delta \vec{\omega}|-0\) and \(\Delta|\vec{\omega}|-0\) (c) \(|\Delta \vec{r}| \neq 0\) but \(\Delta|\vec{r}|-0\) (d) \(|\Delta \vec{a}| \neq 0\) but \(\Delta|\vec{a}|-0\)

Short Answer

Expert verified
Correct options: (a) and (c)

Step by step solution

01

Understanding Uniform Circular Motion

In uniform circular motion, an object moves in a circle at a constant speed. This means that the magnitude of the velocity (speed) remains constant but the direction changes continuously.
02

Evaluation of Option (a)

For option (a), \(|\Delta \vec{v}| eq 0\) indicates that the change in velocity vector is not zero due to the change in direction, but \(\Delta |\vec{v}|=0\) implies that the magnitude of velocity (speed) remains constant. This is true in uniform circular motion since the speed doesn't change but the velocity vector does due to direction change.
03

Evaluation of Option (b)

Option (b) states \(|\Delta \vec{\omega}|=0\) and \(\Delta|\vec{\omega}|=0\), meaning both the change in angular velocity vector and its magnitude are zero. However, in uniform circular motion, the magnitude of angular velocity stays the same, but the angular velocity vector remains constant as direction doesn't change. The statements here are correct for uniform circular motion.
04

Evaluation of Option (c)

For option (c), \(|\Delta \vec{r}| eq 0\) implies that the position vector changes, and \(\Delta|\vec{r}|=0\) means the magnitude of displacement remains constant. This is accurate because the object is moving around a circle, changing position, but the radius (magnitude of position) is constant.
05

Evaluation of Option (d)

Option (d), \(|\Delta \vec{a}| eq 0\) implies that the change in acceleration vector is not zero (due to direction change), but \(\Delta|\vec{a}|=0\) implies the magnitude of the centripetal (radial) acceleration stays constant. This is incorrect in uniform circular motion because centripetal acceleration's magnitude and direction are constant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Velocity Vector
In the context of uniform circular motion, the velocity vector holds a unique role. Though the speed of the object remains unchanged, the velocity vector is constantly changing. This is because velocity is a vector quantity, which means it has both magnitude (speed) and direction. When an object moves in a circle, its direction continuously changes.
  • The magnitude of the velocity (speed) stays constant.
  • The direction of the velocity changes continuously.
  • As a result, the velocity vector is always changing.
This continual change in direction is what results in acceleration, even if the speed of the object doesn鈥檛 change.
Angular Velocity
The concept of angular velocity is crucial in understanding uniform circular motion. Angular velocity (\( \omega \)) measures how quickly an object rotates around a circle.
  • It has both a magnitude and direction, making it a vector.
  • The magnitude of angular velocity stays constant in uniform circular motion.
  • The direction, often represented by the right-hand rule, does not change.
Thus, because there鈥檚 no change in magnitude or direction, the change in angular velocity vector and its magnitude is zero in uniform circular motion.
Centripetal Acceleration
Centripetal acceleration is what keeps an object moving along a circular path. Even when the speed is constant, the direction of motion changes continuously. This change in direction is caused by centripetal acceleration.
  • Centripetal acceleration is directed towards the center of the circle.
  • The magnitude stays the same as long as the speed is constant.
  • Since direction is always towards the center, it remains unchanged.
In the case of uniform circular motion, this acceleration is always present, maintaining the object's path around the circle without altering its speed.
Position Vector
The position vector (\( \vec{r} \)) describes the location of an object in relation to the center of the circle. In circular motion, as the object moves around the circle, its position vector keeps changing its direction.
  • The magnitude of the position vector is the radius of the circle.
  • This magnitude remains constant as it's the same distance from the center.
  • However, the position vector itself changes as the object moves.
Therefore, in uniform circular motion, while the magnitude of the displacement doesn't change, the position vector does because the object鈥檚 direction is always varying along the circle.
Displacement in Circular Motion
Displacement refers to the change in position of the object. In uniform circular motion, analyzing displacement can be insightful.
  • Displacement is a vector with both magnitude and direction.
  • Over an entire revolution, the displacement vector sums to zero as the starting and ending points coincide.
  • However, at any given instance, the direction and magnitude can vary over a short path.
It's important to recognize that though the object鈥檚 path on the circle keeps the radius constant, overall displacement isn't simply zero unless considering full cycles.

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Most popular questions from this chapter

'Ihe equations of motion of a body are given by \(x=36 t\) and \(2 y=96 t-9.8 t^{2}\). (lake \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\) ) (a) 'lhe height of the tower, over which it can just pass (when it is \(100 \mathrm{~m}\) from the point of projection), is \(110.6 \mathrm{~m}\). (b) 'lhe angle of projection is \(\sin ^{-1}(4 / 5)\). (c) 'lhe velocity of projection is \(60 \mathrm{~m} / \mathrm{sec}\). (d) 'lhe range on the horizontal plane is \(200 \mathrm{~m}\)

Two towns \(A\) and \(B\) are connccled by a regular bus service with a bus leaving in cither dircetion cvery \(T\) minutes. A man cycling with a speed of \(20 \mathrm{~km} / \mathrm{h}\) in the dircction \(A\) to \(B\) notices that a bus goes past him cvery \(18 \mathrm{~min}\). in the dircction of his motion and cvery \(6 \mathrm{~min}\). in the opposite dircetion. What is the period \(T\) o \([\) the bus serviec and with what spod (assumed constant) do the buses ply on the road?

The trajectory o a projectile is \(y=a x-b x^{2}\) wherc \(a\) and \(b\) are positive constants find the angle of projection, the maximum height attainod and the horizontal range of the projectile.

A partiele starts from the origin at \(t=0\) with a velocity of \(8.0 \hat{j} \mathrm{~m} / \mathrm{s}\) and moves in the \(x-y\) planc with a constant acecleration of \((4 \hat{i}+2 \hat{j}) \mathrm{m} / \mathrm{s}^{2}\). At the instant the particle's \(x\) -coordinate is \(29 \mathrm{~m}\), what are: (a) its \(y\) -coordinalc and (b) its spced?

\(\Lambda\) particle is found to be at the same point at two instants \(t_{1}\) and \(t_{2}\). Statement-1 : The minimum possiblo value of average speed is \(z\) ero. Statement-2: The average velocity of the particle for the interval \(\left(t_{2}-t_{1}\right)\) is zero.
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